Optimal natural dualities. II: General theory.

A general theory of optimal natural dualities is presented, built on the test algebra technique introduced in an earlier paper. Given that a set $R$ of finitary algebraic relations yields a duality on a class of algebras $\mathcal{A} = \operatorname{\mathbb{I}\mathbb{S}\mathbb{P}}( \underline{M})$, those subsets $R'$ of $R$ which yield optimal dualities are characterised. Further, the manner in which the relations in $R$ are constructed from those in $R'$ is revealed in the important special case that $ \underline{M}$ generates a congruence-distributive variety and is such that each of its subalgebras is subdirectly irreducible. These results are obtained by studying a certain algebraic closure operator, called entailment, definable on any set of algebraic relations on $\underline{M}$. Applied, by way of illustration, to the variety of Kleene algebras and to the proper subvarieties $\mathbf{B}_{n}$ of pseudocomplemented distributive lattices, the theory improves upon and illuminates previous results. [ABSTRACT FROM AUTHOR]